Tuesday, February 27, 2018

High School -- Geometry Performance Task: Properties of Quadrilaterals (Day 117)

Today I subbed in another math class -- at a different high school from yesterday. This time I subbed in a Geometry class. I definitely want to do the "Day in the Life" for today, since this is, after all, a Geometry blog.

And so here is the "Day in the Life" for today -- Day 109 in this district:

7:00 -- Rise and shine -- this is yet another "first period" (zero period) class.

The Geometry students are still working on similarity -- Lesson 7-3 of the Glencoe text. In fact, this is what I wrote three years ago about today's lesson (from a tutor's perspective):

Last night I tutored my geometry student again. Section 7-3 of the Glencoe text is on similar triangles, including all three similarity statements: AA, SAS, and SSS.

Well, my student is still slightly confused with solving and setting up proportions. But he was more concerned with a question from Section 7-2 of the Glencoe text, on similar polygons. There is a group of similarity problems in the Glencoe text that are flawed.

Three years ago I ended up writing more about Lesson 7-2 than 7-3. But unfortunately, one of the questions on today's Lesson 7-3 worksheet also contains a (different) flaw, as I'll soon find out.

Most of the first few problems have the students calculate missing lengths in similar triangles. Notice that since one of the side lengths is missing, SSS Similarity goes right out the door. Thus all of the problems on the first side of the worksheet use either AA~ or SAS~.

7:55 -- First period leaves and second period begins. They begin the Lesson 7-3 worksheet.

8:50 -- Second period leaves and third period begins. They begin the Lesson 7-3 worksheet.

9:45 -- This is the same school I subbed at a week ago today -- so now it's tutorial time. Several students arrive for tutorial, but many of them are working on subjects other than math. The two Geometry students I see have a different teacher and are working in a different lesson. I do help these two briefly.

10:25 -- Tutorial ends -- and believe it or not, so does my school day! The teacher I'm subbing for isn't merely a coach -- he's the athletic director. And so in order to allow him to organize and plan all the school sports, he's given only a three-class load that ends at nutrition time.

So this is the shortest "Day in the Life" I've ever done. But I still have plenty to say, about both the curriculum and my New Year's Resolutions.

Yesterday I covered the first resolution, but my only chance to follow it was during sixth period. As this is such an important resolution, let's continue to focus on Resolution #1 for today's half-day:

1. Implement a classroom management system based on how students actually think.

It's often tricky to implement classroom management when students are working on different assignments or finishing a multi-day assignment. Students can always claim "I'm done, so let me do whatever I want."

Today, on the other hand, everyone is working on the Lesson 7-3 worksheet. This means that I can manage the classroom strongly -- just make sure that everyone is working on it.

When first period begins, I do problem #1 on the board. Later on, I do #2 as well. I'm hoping that the students can finish at least the first side of the paper. But many students fail to finish -- and those who do the work skip questions or jump to the other side. I end up having to leave the names of two students who do nothing other than copy #1 and #2 from the board.

In second period, I leave Questions #1 and #2 on the board and start on Question #3. Once I try the problem, it's no longer any wonder why students keep skipping it -- the question is flawed.

Here is the Question #3 from the Glencoe Lesson 7-3 worksheet:

3. Identify the similar triangles. Then find the measure of QR.

In Triangle QSV, point R is between Q and S, while T is between V and S. The following lengths are given on the diagram:

QS = VS = 36
QV = 36sqrt(2)
TS = 24
QR = x

I tell my students another story from my tutoring days, which I've also chronicled on the blog:

I do admit that students might be tricked by this sort of question. I remember once when I was teaching or tutoring a student who was in the similarity chapter of the text. Upon seeing the two triangles in a question much like this one, he immediately started writing a proportion. I told him that a proportion can be set up only if the triangles are similar, and so I asked him, how did he know that the triangles are similar? His response was, of course the triangles were similar because we were in Chapter 7, the similarity chapter of the text!

Yes, my student cleverly figured out that in the similarity chapter, nearly every problem would have a pair of similar triangles. But this problem illustrates what's wrong with his thinking

And here's the problem on the worksheet with today's students -- the given info isn't sufficient to conclude that the triangles are similar!

If you don't believe me, then try to prove that Triangle QSV ~ RST. We've already eliminated SSS~ as a possibility for any question on this worksheet, but what about SAS~? The two triangles have Angle S in common, but one of the adjacent sides, RS, has length 36 - x. Since the unknown value that we;re solving for is part of one of the sides, this rules out SAS~ as a possibility as well.

This leaves only AA~. But no angle information is given, so we cannot conclude that Angles Q and TRS are congruent, nor Angles V and RTS. Notice that in the next three questions, we're given that two of the sides are parallel (which would give us congruent corresponding angles), but no parallel lines are stated in this question.

Notice that if we could conclude that Triangle QSV ~ RST, then x = 12. But what's to stop us from choosing another point on QS, calling it R', and then claiming that QR' = 12 instead of QR?

Ironically, here are the instructions for the first four questions on the other side of the worksheet:

"Determine whether each pair of triangles is similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Explain your reasoning."

And doubly ironically, all four pairs of triangles have enough info to conclude they're similar. (There are two SAS~ questions and one each of SSS~ and AA~.) The only pair of triangles not to contain sufficient info is on the wrong side of the worksheet as #3.

Well, let's answer this question for the wrong #3. Here is what we need to conclude that Triangles QSV and RST are indeed similar. Any one of the following is sufficient:

  • Angle Q = TRS
  • Angle V = RTS
  • QV | | RT
  • RT = 24sqrt(2)
This last one needs some explanation. The side lengths of QSV are 36, 36, 36sqrt(2). This should be recognizable as the sides of a 45-45-90 triangle, with the right angle at S. Then knowing RT will allow us to conclude that the triangles are similar by HL. For some strange reason, HL Similarity never appears in any texts, yet it's obvious that HL~ should work. The same proof for the other similarity theorems works for HL~ as well.

There's one more sufficient statement that will allow us to conclude similarity -- RS = 24, then the triangles are similar by SAS~. But if we had RS, then we could find x = QR by the Betweenness Theorem (Segment Addition) as QR = QS - RS, without using similarity at all.

In second period, I end up spending so much time on Question #3 that I rush through the final problem on the front of the worksheet -- Question #7, a word problem on shadow lengths. Students, of course, try to avoid word problems, and so many don't finish Question #7. (Some students also try to avoid Question #3 because of the square root, which isn't the real problem with it.)

But returning to classroom management, I do catch one student who doesn't even have his worksheet on his desk. I can't catch his name, so I must return to identifying him by backpack. Since it's still wintertime, he's wearing a coat, which I also use for identification. Students wear different clothes everyday, but they often wear the same jacket -- and of course, they don't change their backpack.

In third period, I do Questions #1, #2, and then #4-7 on the board, saving a discussion about Question #3 for last. Students still have some trouble with the word problem in #7, but this time I'm able to get them through it. I don't need to write any names.

During tutorial, another management issue turns up. This is one that has come up with several other teachers I've spoken to during breaks. A student texts someone on a cell phone, then claims that the recipient is her mother about to undergo surgery. Indeed, this is what the other sub I saw in the lounge had to kick out a student for yesterday. That student yelled at the sub that some things, like life and death, are more important than school, and that exceptions to the no phone rule should always, always, be granted during emergencies, no questions asked.

From my perspective, this is a blatant attempt to neuter the no phone rule. Students who claim that they are texting a parent are almost always texting another teenager, and the topic isn't an emergency, but entertainment. Their idea is that since teachers can never be certain that a text isn't an emergency, the only 100% fair thing to do is to let students text whenever they want to whomever they want, even if it interferes with their education. To them, letting kids do what they want is the only fair thing, and everything else is unfair. To them, if you want to be considered a "good" or "fair" teacher, it's very, very important to let them have fun, to the exclusion of everything else.

And returning to the girl in my classroom, the second time she takes out her phone, she and a group of girls started laughing -- odd behavior for someone whose mom is undergoing surgery. This clinches it, of course -- she's texting another teen to discuss entertainment. The problem, of course, is that this is tutorial, so it's impossible to get any identifying information for a student who might not even be in any of his classes. Of course, she claims that she's finished all her written work. In other words, to her, the only fair thing to do is let her have a free 40-minute texting and entertainment period everyday (since she can claim she has no work everyday) -- nothing less is fair.

Last year at my middle school, only three students claimed that emergencies should override the no phone rule. Two of the claims turned out to be genuine -- one was the special scholar (January 6th post) whose mother was making arrangements to pick up her elementary school sister, and the other had a death in her family. (She and her younger brother were eventually picked up from school.) To this day, I still believe the third student was lying about the emergency.

If I were a parent, I would never text my children during the school day, not even if a death in the family happens. If a relative were to die one minute after school starts, I would wait seven full hours, until one minute after the final dismissal bell, to text my children. I'd be too shocked by the news even to consider texting anyone. And besides, I'd assume that my kids would have their phones confiscated if they were to text -- the last thing they need is to lose a relative and a phone the same day. If it's absolutely necessary to contact them, I'd make every effort to do so without texting them -- including contacting or driving up to the school. And if I can't find any way to contact them without a text, I'd tell them not to respond until after the final dismissal bell. And if I were a student, I might not even want to text my parents anyway! (Note: with shootings in the news, here I assume that if an active shooter arrives at my kids' school, all classes are cancelled and students can safely text without phone confiscation.)

OK, that's enough about classroom management for today. I do wish to write a little more about teaching the Glencoe text. If I were a regular teacher with the Glencoe text, how would I pace it.

Since the Glencoe text has a Chapter 0, it's tempting to follow the same digit pattern that I've established for the U of Chicago text. But this is a little too fast -- it forces Chapter 7 on similarity into the first semester. I like the idea of starting the second semester with similarity -- and apparently this is exactly what's done in this district.

Of course, to be still in Chapter 7 five weeks into the semester is a bit slow, especially if the goal is to reach Chapter 13 by the end of the year. A better pace would be to start the Big March with Chapter 8, on trigonometry. (That's right -- trig at the Big March again!) Apparently, the Geometry students I see in tutorial are already in Chapter 8. They are studying special right triangles -- and they are able to recognize the 36-36-36sqrt(2) still written on the board as a special 45-45-90 triangle.

In the Glencoe text, Chapter 4, on congruence, appears before Chapter 9, on transformations. I find an old copy of the first semester final in the classroom, and apparently, Chapter 9 is pushed up into the first semester. This fits the Common Core definition of congruence. And it also appears that Lesson 9-6 is combined with Chapter 7. This makes sense as Lesson 9-6 is on dilations.

I also find a copy of a Performance Task, similar to what students may find on the SBAC. There's a hole in my U of Chicago pacing plan since Chapter 11 has only six sections. And I've never posted a Performance Task before, despite this being a Common Core blog.

And so I post this task as an activity for today and tomorrow. Actually, I create my own version of the problem rather than the district copy. This is to block out the name of the district since I don't post identifying information (and to avoid any issues with SBAC, which might mistake this practice question for a real test question). Also, I changed it from one day to two days. On the actual SBAC, students are typically given a two-hour block to complete the Performance Task, so they should have two days to complete it.

The question is about the coordinates of a square. It fits perfectly with Chapter 11 of the U of Chicago text -- and indeed it's similar to the Lesson 11-1 activity from two weeks ago.

In the Glencoe text, the Performance Task fits with Chapter 6, on quadrilaterals. Glencoe Chapter 6 is similar to Chapter 5 in the U of Chicago text (and Chapter 6 in the Third Edition), except that coordinates appear early. Indeed, I lamented three years ago that the Distance Formula appears as early as Chapter 1 in the Glencoe text!

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